SEARCH FOR INSPIRALING BINARIES

S. V. Dhurandhar

IUCAA

Pune, India

Inspiraling compact binaries

GW

The inspiraling compact binary

• Two compact objects: Neutron stars /Blackholes spiraling in together

• Waveform cleanly modeled

3/213/5

23

100100105.2~

Hz

f

Mpc

rh a

sunMM

Parameter Space for 1 – 30 solar masses

Hzfa 40

The matched filter

)(

)()(

)()()(

~~

fS

fhfq

dttqtxc

h

Optimal filter in Gaussian noise:

Detection probability is maximised for a given false alarm rate

Stationary noise:

Matched filtering the inspiraling binary signal

Detection Strategy: Maximum Likelihood

Parameters: Amplitude, ta , a , m1 , m2

• Amplitude: Use normalised templates

• Scanning over ta : FFT

• Initial phase a : Two templates for 0 and and add in

quadrature

• masses chirp times: bank required

Signal depends on many parameters

Detection

Data: x (t)

Templates: h0 (t; ) and h(t; )

Padding: about 4 times the longest signal

Fix

Outputs: Use FFT to compute c0 (and c(

is the time-lag: scan over ta

Statistic: c2(c02

(+ c(Maximised over

initial phase

Conventionally we use its square root c(

Compare c( with the threshold

FLAT SEARCH

Parameter space for the mass range 1 – 30 solar masses

Hzfa 40

Parameter Space

Hexagonal tiling of the parameter space

Minimal match: 0 .97 Density of templates: ~ 1300/sec2

LIGO I psd

Cost of the flat search

Longest chirp: 95 secs for lower cut-off of 30 Hz

Sampling rate: 2048 Hz

Length of data train (> x 4): 512 secs

Number of points in the data train: N = 220

Length of padding of zeros: 512 – 95 = 417 secs

This is also the processed length of data

Number of templates: nt = 11050

Online speed reqd. = nt x 6 N log2 N / 417

~ 3.3 GFlop

Hierarchical Search

Comparison of contours (H = 0.8) with 256 Hz and 1024 Hz cut-off

Only hierarchy in masses: gain factor 25-30

Boundary effects not included

Hierarchy on masses only

92% power at

fc = 256 Hz

Choice of the first stage threshold

• High enough to reduce false alarm rate

-- reduce cost in the second stage

-- For 218 points in a data train or so

• Low enough to make the 1st stage bank as coarse as possible and hence reduce the cost in the 1st stage

Smin)S ~ 6

The first stage templates

Templates flowing out of the deemed parameter space are not wasted

Zoomed view near low mass end

Need 535 templates

Cost of the EHS Longest chirp ~ 95 secs Tdata = 512 secs Tpad = 417 secs

f 1samp = 512 Hz f 2

samp = 2048 Hz

N 1 = 2 18 N 2 = 2 20

= .92 Smin S = 8.2

~ 6.05

n1 = 535 templates n2 = 11050 templates

Smin ~ 9.2

S1 ~ 6 n1 N 1 log2 N 1 / Tpad ~ 36 Mflops

S2 ~ ncross 6 N 2 log2 N 2 / Tpad ~ 13 Mflops

S ~ 49 Mflops Sflat ~ 3.3 GFlops

ncross ~ .82

Gain ~ 68

Optimised EHS

Performance of the code on real data

• S2 - L1 playground data was used – 256 seconds chunks, segwizard – data quality flags

• 18 hardware injections - recovered impulse time and SNR – matched filtering part of code was validated

• Monte-Carlo chirp injections with constant SNR - 679 chunks

EHS pipeline ran successfully

• The coarse bank level generates many triggers which pass the and thresholds

Clustering algorithm

The EHS pipeline

Can run in standalone mode (local Beowulf) or in LDAS

Data conditioning

Preliminary data conditioning: LDAS datacondAPI

• Downsamples the data at 1024 and 4096 Hz

• Welch psd estimate

• Reference calibration is provided

Code data conditioning:

• Calculate response function, psd, h(f)

• Inject a signal

Template banks and matched filtering

• LAL routines generate fine and the coarse bank

• Compute for templates with initial phases of 0 and

• Hundreds of events are generated and hence a clustering algorithm is needed

The Clustering Algorithm

Step 1: DBScan algorithm is used to identify clusters/islands

Step 2: The cluster is condensed into an event which is followed up by a local fine bank search

DBScan (Ester et al 1996)

1. For every point p in C there is a q in C such that p is in the -nbhd of q

This breaks up the set of triggers into components

2. A cluster must have at least Nmin number of points – otherwise its an outlier

Condensing a cluster

x

xdxl

dxxlxxmean

)(

)(

k k

k

nm

2

1

Figure of merit:

l(x) is the Lagrange interpolation polynomial

Conclusions

• Galaxy based Monte Carlo simulation is required to figure out the efficiency of detection. This will incorporate current understanding of BNS distribution in masses and in space

• Hierarchical search frees up CPU for additional parameters

CONCLUSION

* Results shown for Gaussian noise

- Now looking at real data – S1, S2

- Performance in non-Gaussian noise

- Event multiplicity needs to be condensed into a single event

* Reducing computational cost with hierarchical approach frees up CPU power for additional parameters